18.175: Lecture 23 Random walks Scott Sheffield MIT 18.175 Lecture 23 Outline Random walks Stopping times Arcsin law, other SRW stories Outline Random walks Stopping times Arcsin law, other SRW stories Exchangeable events I Start with measure space (S, S, µ). Let Ω = {(ω1 , ω2 , . . .) : ωi ∈ S}, let F be product σ-algebra and P the product probability measure. 18.175 Lecture 23 Exchangeable events I Start with measure space (S, S, µ). Let Ω = {(ω1 , ω2 , . . .) : ωi ∈ S}, let F be product σ-algebra and P the product probability measure. I Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. 18.175 Lecture 23 Exchangeable events I Start with measure space (S, S, µ). Let Ω = {(ω1 , ω2 , . . .) : ωi ∈ S}, let F be product σ-algebra and P the product probability measure. I Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. I Event A ∈ F is permutable if it is invariant under any finite permutation of the ωi . 18.175 Lecture 23 Exchangeable events I Start with measure space (S, S, µ). Let Ω = {(ω1 , ω2 , . . .) : ωi ∈ S}, let F be product σ-algebra and P the product probability measure. I Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. I Event A ∈ F is permutable if it is invariant under any finite permutation of the ωi . I Let E be the σ-field of permutable events. 18.175 Lecture 23 Exchangeable events I Start with measure space (S, S, µ). Let Ω = {(ω1 , ω2 , . . .) : ωi ∈ S}, let F be product σ-algebra and P the product probability measure. I Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. I Event A ∈ F is permutable if it is invariant under any finite permutation of the ωi . I Let E be the σ-field of permutable events. I This is related to the tail σ-algebra we introduced earlier in the course. Bigger or smaller? 18.175 Lecture 23 Hewitt-Savage 0-1 law I If X1 , X2 , . . . are i.i.d. and A ∈ A then P(A) ∈ {0, 1}. Hewitt-Savage 0-1 law I If X1 , X2 , . . . are i.i.d. and A ∈ A then P(A) ∈ {0, 1}. I Idea of proof: Try to show A is independent of itself, i.e., that P(A) = P(A ∩ A) = P(A)P(A). Start with measure theoretic fact that we can approximate A by a set An in σ-algebra generated by X1 , . . . Xn , so that symmetric difference of A and An has very small probability. Note that An is independent of event A0n that An holds when X1 , . . . , Xn and Xn1 , . . . , X2n are swapped. Symmetric difference between A and A0n is also small, so A is independent of itself up to this small error. Then make error arbitrarily small. 18.175 Lecture 23 Application of Hewitt-Savage: I If Xi are i.i.d. in Rn then Sn = Rn . 18.175 Lecture 23 Pn i=1 Xi is a random walk on Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: 18.175 Lecture 23 Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: I Sn = 0 for all n 18.175 Lecture 23 Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: I I Sn = 0 for all n Sn → ∞ 18.175 Lecture 23 Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: I I I Sn = 0 for all n Sn → ∞ Sn → −∞ 18.175 Lecture 23 Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: I I I I Sn = 0 for all n Sn → ∞ Sn → −∞ −∞ = lim inf Sn < lim sup Sn = ∞ 18.175 Lecture 23 Application of Hewitt-Savage: I I P If Xi are i.i.d. in Rn then Sn = ni=1 Xi is a random walk on Rn . Theorem: if Sn is a random walk on R then one of the following occurs with probability one: I I I I I Sn = 0 for all n Sn → ∞ Sn → −∞ −∞ = lim inf Sn < lim sup Sn = ∞ Idea of proof: Hewitt-Savage implies the lim sup Sn and lim inf Sn are almost sure constants in [−∞, ∞]. Note that if X1 is not a.s. constant, then both values would depend on X1 if they were not in ±∞ 18.175 Lecture 23 Outline Random walks Stopping times Arcsin law, other SRW stories Outline Random walks Stopping times Arcsin law, other SRW stories Stopping time definition I Say that T is a stopping time if the event that T = n is in Fn for i ≤ n. Stopping time definition I Say that T is a stopping time if the event that T = n is in Fn for i ≤ n. I In finance applications, T might be the time one sells a stock. Then this states that the decision to sell at time n depends only on prices up to time n, not on (as yet unknown) future prices. Stopping time examples I Let A1 , . . . be i.i.d. random variables equal to −1 with probability P .5 and 1 with probability .5 and let X0 = 0 and Xn = ni=1 Ai for n ≥ 0. 18.175 Lecture 23 Stopping time examples I I Let A1 , . . . be i.i.d. random variables equal to −1 with probability P .5 and 1 with probability .5 and let X0 = 0 and Xn = ni=1 Ai for n ≥ 0. Which of the following is a stopping time? The smallest T for which |XT | = 50 The smallest T for which XT ∈ {−10, 100} The smallest T for which XT = 0. The T at which the Xn sequence achieves the value 17 for the 9th time. 5. The value of T ∈ {0, 1, 2, . . . , 100} for which XT is largest. 6. The largest T ∈ {0, 1, 2, . . . , 100} for which XT = 0. 1. 2. 3. 4. 18.175 Lecture 23 Stopping time examples I I Let A1 , . . . be i.i.d. random variables equal to −1 with probability P .5 and 1 with probability .5 and let X0 = 0 and Xn = ni=1 Ai for n ≥ 0. Which of the following is a stopping time? The smallest T for which |XT | = 50 The smallest T for which XT ∈ {−10, 100} The smallest T for which XT = 0. The T at which the Xn sequence achieves the value 17 for the 9th time. 5. The value of T ∈ {0, 1, 2, . . . , 100} for which XT is largest. 6. The largest T ∈ {0, 1, 2, . . . , 100} for which XT = 0. 1. 2. 3. 4. 18.175 Lecture 23 Stopping time examples I I Let A1 , . . . be i.i.d. random variables equal to −1 with probability P .5 and 1 with probability .5 and let X0 = 0 and Xn = ni=1 Ai for n ≥ 0. Which of the following is a stopping time? The smallest T for which |XT | = 50 The smallest T for which XT ∈ {−10, 100} The smallest T for which XT = 0. The T at which the Xn sequence achieves the value 17 for the 9th time. 5. The value of T ∈ {0, 1, 2, . . . , 100} for which XT is largest. 6. The largest T ∈ {0, 1, 2, . . . , 100} for which XT = 0. 1. 2. 3. 4. I Answer: first four, not last two. 18.175 Lecture 23 Stopping time theorems I Theorem: Let X1 , X2 , . . . be i.i.d. and N a stopping time with N < ∞. 18.175 Lecture 23 Stopping time theorems I Theorem: Let X1 , X2 , . . . be i.i.d. and N a stopping time with N < ∞. I Conditioned on stopping time N < ∞, conditional law of {XN+n , n ≥ 1} is independent of Fn and has same law as original sequence. 18.175 Lecture 23 Stopping time theorems I Theorem: Let X1 , X2 , . . . be i.i.d. and N a stopping time with N < ∞. I Conditioned on stopping time N < ∞, conditional law of {XN+n , n ≥ 1} is independent of Fn and has same law as original sequence. I Wald’s equation: Let Xi be i.i.d. with E |Xi | < ∞. If N is a stopping time with EN < ∞ then ESN = EX1 EN. 18.175 Lecture 23 Stopping time theorems I Theorem: Let X1 , X2 , . . . be i.i.d. and N a stopping time with N < ∞. I Conditioned on stopping time N < ∞, conditional law of {XN+n , n ≥ 1} is independent of Fn and has same law as original sequence. I Wald’s equation: Let Xi be i.i.d. with E |Xi | < ∞. If N is a stopping time with EN < ∞ then ESN = EX1 EN. I Wald’s second equation: Let Xi be i.i.d. with E |Xi | = 0 and EXi2 = σ 2 < ∞. If N is a stopping time with EN < ∞ then ESN = σ 2 EN. 18.175 Lecture 23 Wald applications to SRW I S0 = a ∈ Z and at each time step Sj independently changes by ±1 according to a fair coin toss. Fix A ∈ Z and let N = inf{k : Sk ∈ {0, A}. What is ESN ? 18.175 Lecture 23 Wald applications to SRW I S0 = a ∈ Z and at each time step Sj independently changes by ±1 according to a fair coin toss. Fix A ∈ Z and let N = inf{k : Sk ∈ {0, A}. What is ESN ? I What is EN? 18.175 Lecture 23 Outline Random walks Stopping times Arcsin law, other SRW stories Outline Random walks Stopping times Arcsin law, other SRW stories Reflection principle I How many walks from (0, x) to (n, y ) that don’t cross the horizontal axis? Reflection principle I How many walks from (0, x) to (n, y ) that don’t cross the horizontal axis? I Try counting walks that do cross by giving bijection to walks from (0, −x) to (n, y ). Ballot Theorem I Suppose that in election candidate A gets α votes and B gets β < α votes. What’s probability that A is a head throughout the counting? 18.175 Lecture 23 Ballot Theorem I Suppose that in election candidate A gets α votes and B gets β < α votes. What’s probability that A is a head throughout the counting? I Answer: (α − β)/(α + β). Can be proved using reflection principle. 18.175 Lecture 23 Arcsin theorem I Theorem for last hitting time. 18.175 Lecture 23 Arcsin theorem I Theorem for last hitting time. I Theorem for amount of positive positive time. 18.175 Lecture 23